Aug 2026· 1 citation· ⚡ 1 influential· 16 references
Mathematics
Abstract
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric Chow-type characterization $$X\text{ is algebraic} \quad \Longleftrightarrow \quad \mathcal{H}^{2d} \bigl( \Sigma_\infty(X)\bigr)=0. $$ As an application, if $C_\infty(X)$ is definable in an o-minimal expansion of $(\mathbb R,+,\cdot)$ and $\dim_{\mathbb R} C_\infty(X)\le 2d$, then $X$ is algebraic.
Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\tau} \text{ for infinitely many }(\mathbf{p},q)\in\mathbb{Z}^d\times\mathbb{N} \right\}. \] We prove that there exists $\varepsilon_K>0$ such that, for every $1/d<\tau<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(\tau)}(K\cap W_d(\tau))=\infty, \qquad\text{with } s(\tau):=\delta+\frac{d+1}{1+\tau}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(\tau)) = \delta+\frac{d+1}{1+\tau}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud--Durand conjectural formula for $\tau>1$ sufficiently close to $1$.
Let $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\varGamma=\pi_1(\Sigma)$, let $\rho_M:\varGamma\longrightarrow \mathrm{PSL}_2(\mathbb{C})$ be the holonomy of $M$ and $\rho:\varGamma\longrightarrow \mathrm{PSL}_2(\mathbb{C})$ any representation in $\mathrm{PSL}_2(\mathbb{C})$. We will show that, if $\rho$ is discrete and non-faithful, then \[ \|[\mathrm{Vol}(\rho)]-[\mathrm{Vol}(\rho_M)]\|_\infty\geq \boldsymbol{v}_3 \] holds, where $[\mathrm{Vol}(\rho)]$ denotes the bounded fundamental class of $\rho$ in the bounded cohomology $H_b^3(\varGamma,\mathbb{R})$ of $\varGamma$ and $\boldsymbol{v}_3$ is the volume of a regular ideal 3-simplex in $\mathbb{H}^3$. As an application, we present a rigidity theorem for $\rho_M$ in the set of representations $\rho$ of $\varGamma$ in $\mathrm{PSL}_2(\mathbb{C})$ in terms of $[\mathrm{Vol}(\rho)]$. The rigidity theorem implies that $\boldsymbol{v}_3$ is the optimal separating constant.
We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\nabla\Phi$ be the Brenier map from $\mu$ to $\nu$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla\Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier $\mathfrak{m}_{\sqrt{\lfloor t^2\rfloor}}(\xi)$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $\xi$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wr\'obel \cite{MSW25}.
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert>t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.
Let $F$ and $E$ be finite extensions of $\mathbb Q_p$, let $\mathbb G$ be a reductive group over $F$, and put $G=\mathbb G(F)$. Let $V$ be a locally algebraic representation of the form $V=\pi_{\mathrm{sm}}\otimes_E\sigma_{\mathrm{alg}}$, where $\pi_{\mathrm{sm}}$ is smooth admissible and $\sigma_{\mathrm{alg}}$ is finite-dimensional algebraic. We study the extended Goldman--Iwahori distance on the set of non-Archimedean norms on $V$. After fixing a reference norm $\alpha_0$, its finite-distance component $\mathscr N_{\alpha_0}(V)$ is the bounded projective limit of the extended Bruhat--Tits buildings attached to $V_K=\pi_{\mathrm{sm}}^K\otimes_E\sigma_{\mathrm{alg}}$. It is complete for the resulting uniform sup metric; this metric is of $\ell^\infty$ type and is generally not CAT(0). We prove directly that a $G$-orbit in $\mathscr N_{\alpha_0}(V)$ is bounded if and only if this component contains a $G$-invariant norm. The invariant norm is the pointwise supremum of the orbit. We formulate an integral group-algebra and type-Hecke condition necessary for an invariant norm. For $G=GL_n(F)$ we specialise to $V=\operatorname{BS}(r)=\pi_{\mathrm{gen}}(r)\otimes_E \pi_{\mathrm{alg}}(r)$.
A. Pyvovarov· 0 citations
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